Octic Hilbert 2-class fields of real quadratic fields with discriminant 8p

Abstract

In this article we explain how to construct cyclic octic unramfied extensions of the real quadratic number field $k = {\mathbb Q}(\sqrt{2p}\,)$, where $p \equiv 1 \bmod 8$ is a prime number such that $h_2(k) \equiv 0 \bmod 8$. The construction only requires solving the diophantine equation $eu^2 = t^2 + 2ps^2$ in integers.